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Time reversal (reversal of motion) The fact that the operator K does not change the base states can be justified like: The state |a0 i represented in the base {|a0 i} maps to the column vector 0 0 then also x(−t) is a solution. .. . At the moment t = 0 let there be a particle at the point 0 0 x(t = 0) with the momentum p(t = 0). Then a particle at |a i 7→ 1 , the same point but with the momentum −p(t = 0) 0 follows the trajectory x(−t). . In the quantum mechanical Schrödinger equation .. 0 h̄2 2 ∂ψ = − ∇ + V ψ, ih̄ which is unaffected by the complex conjugation. ∂t 2m Note The effect of the operator K depends thus on the due to the first derivative with respect to the time, choice of the basis states. ψ(x, −t) is not a solution eventhough ψ(x, t) were, but If U is a unitary operator then the operator θ = U K is ψ ∗ (x, −t) is. In quantum mechanics the time reversal has antiunitary. obviously something to do with the complex conjugation. Proof: Firstly Let us consider the symmetry operation θ(c1 |αi + c2 |βi) = U K(c1 |αi + c2 |βi) |αi −→ |α̃i, |βi −→ |β̃i. = (c∗1 U K|αi + c∗2 U K|βi) = (c∗1 θ|αi + c∗2 θ|βi), We require that the absolute value of the scalar product is invariant under that operation: so θ is antiliniear. Secondly, expanding the states |αi and |βi in a complete basis {|a0 i} we get |hβ̃|α̃i| = |hβ|αi|. X θ ha0 |αi∗ U K|a0 i |αi −→ |α̃i = There are two possibilities to satisfy this condition: The Newton equations of motion are invariant under the transformation t −→ −t: if x(t) is a solution of the equation mẍ = −∇V (x) a0 1. hβ̃|α̃i = hβ|αi, so the corresponding symmetry operator is unitary, that is hβ|αi −→ hβ|U U |αi = hβ|αi. 2. hβ̃|α̃i = hβ|αi∗ = hα|βi, so the symmetry operator cannot be unitary. We define the antiunitary operator θ so that hβ̃|α̃i = hα|βi∗ θ(c1 |αi + c2 |βi) = c∗1 θ|αi + c∗2 θ|βi, where = X 0 ha |αi∗ U |a0 i a0 † The symmetries treated earlier have obeyed this condition. = X hα|a0 iU |a0 i a0 and |β̃i = X ha0 |βi∗ U |a0 i ↔ hβ̃| = a0 X ha0 |βiha0 |U † . a0 Thus the scalar product is XX ha00 |βiha00 |U † U |a0 ihα|a0 i hβ̃|α̃i = a00 = X a0 hα|a0 iha0 |βi = hα|βi a0 = hβ|αi∗ . |αi −→ |α̃i = θ|αi, |βi −→ |β̃i = θ|βi If the operator satisfies only the last condition it is called antilinear. We define the complex conjugation operator K so that where Θ|αi is the time reversed (motion reversed) state. If |αi is the momentum eigenstate |p0 i, we should have Kc|αi = c∗ K|αi. We present the state |αi in the base {|a0 i}. The effect of the operator K is then X X K |αi = |a0 iha0 |αi −→ |α̃i = ha0 |αi∗ K|a0 i a0 = X a0 a0 0 ∗ 0 ha |αi |a i. The operator θ is thus indeed antiunitary. Let Θ be the time reversal operator. We consider the transformation |αi −→ Θ|αi, Θ|p0 i = eiϕ | − p0 i. Let the system be at the moment t = 0 in the state |αi. At a slightly later moment t = δt it is in the state iH |α, t0 = 0; t = δti = 1 − δt |αi. h̄ We apply now, at the moment t = 0, the time reversal operator Θ and let the system evolve under the Hamiltonian H. Then at the moment δt the system is in the state iH δt Θ|αi. 1− h̄ and If the motion of the system is invariant under time reversal this state should be the same as In partcular, for a Hermitean observable A we have Θ|α, t0 = 0; −δti, i.e. we first look at the state at the earlier moment −δt and then reverse the direction of the momentum p. Mathematically this condition can be expressed as iH iH 1− δt Θ|αi = Θ 1 − (−δt) |αi. h̄ h̄ Thus we must have −iHΘ|i = ΘiH|i, where |i stands for an arbitrary state vector. If Θ were linear we would obtain the anticommutator relation HΘ = −ΘH. If now |ni is an energy eigenstate corresponding to the eigenvalue En then, according to the anticommutation rule HΘ|ni = −ΘH|ni = (−En )Θ|ni, and the state Θ|ni is an energy eigenstate corresponding to the eigenvalue −En . Thus most systems (those, whose energy spectrum is not bounded) would not have any ground state. Thus the operator Θ must be antilinear, and, in order to be a symmetry operator, it must be antiunitary. Using the antilinearity for the right hand side of the condition hβ| ⊗ |αi = hγ|αi = hα̃|γ̃i = hα̃|Θ ⊗† |βi = hα̃|Θ ⊗† Θ−1 Θ|βi = hα̃|Θ ⊗† Θ−1 |β̃i. hβ|A|αi = hα̃|ΘAΘ−1 |β̃i. We say that the observable A is even or odd under time reversal depending on wheter in the equation ΘAΘ−1 = ±A the upper or the lower sign holds. This together with the equation hβ|A|αi = hα̃|ΘAΘ−1 |β̃i imposes certain conditions on the phases of the matrix elements of the operator A between the time reversed states. Namely, they has to satisfy hβ|A|αi = ±hβ̃|A|α̃i∗ . In particular, the expectation value satisfies the condition hα|A|αi = ±hα̃|A|α̃i. Example The expectation value of the momentum operator p. We require that hα|p|αi = −hα̃|p|α̃i, so p is odd, or ΘpΘ−1 = −p. The momentum eigenstates satisfy pΘ|p0 i = −ΘpΘ−1 Θ|p0 i = (−p0 )Θ|p0 i, −iHΘ|i = ΘiH|i i.e. Θ|p0 i is the momentum eigenstates correponding to the eigenvalue −p0 : we can write it as ΘiH|i = −iΘH|i. So, we see that the operators commute: ΘH = HΘ. Θ|p0 i = eiϕ | − p0 i. Similarly we can derive for the position operator x the expressions ΘxΘ−1 = x Θ|x0 i = |x0 i Note We have not defined the Hermitean conjugate of the antiunitary operator θ nor have we defined the meaning of the expression hβ|θ. That being, we let the time reversal operator Θ to operate always on the right and with the matrix element hβ|Θ|αi we mean the expression (hβ|) · (Θ|αi). Let ⊗ be an arbitrary linear operator. We define when we impose the physically sensible condition hα|x|αi = hα̃|x|α̃i. We consider the basic commutation relations |γi ≡ ⊗† |βi, so that [xi , pj ]|i = ih̄δij |i. Now hβ|⊗ = hγ| Θ[xi , pj ]Θ−1 Θ|i = Θih̄δij |i, from which, using the antilinearity and the time reversal properties of the operators x and p we get [xi , (−pj )]Θ|i = −ih̄δij Θ|i. We see thus that the commutation rule [xi , pj ]|i = ih̄δij |i remains invariant under the time reversal. Correspondingly, the requirement of the invariance of the commutation rule [Ji , Jj ] = ih̄ijk Jk leads to the condition ΘJ Θ−1 = −J . This agrees with transformation properties of the orbital angular momentum x × p. Wave functions so the states |ni and Θ|ni have the same energy. Because the state |ni was supposed to be nondegenerate they must represent the same state. The wave function of the state |ni is hx0 |ni and the one of the state Θ|ni correspondingly hx0 |ni∗ . These must be same (or more accurately, they can differ only by a phase factor which does not depend on the coordinate x0 ), i.e. hx0 |ni = hx0 |ni∗ For example the wave function of a nondegenerate groundstate is always real. For a spinles particle in the state |αi we get Z Θ|αi = Θ dx0 hx0 |αi|x0 i Z = dx0 hx0 |αi∗ |x0 i = K|αi, Z d3 x0 Θ|x0 ihx0 |αi∗ i.e. the time reversal is equivalent to the complex conjugation. On the other hand, in the momentum space we have Z Θ|αi = d3 p0 | − p0 ihp0 |αi∗ Z = d3 p0 |p0 ih−p0 |αi∗ , Z d3 x0 |x0 ihx0 |αi∗ , because We expand the state |αi with the help of position eigenstates: Z |αi = d3 x0 |x0 ihx0 |αi. Now Θ|αi = = HΘ|ni = ΘH|ni = En Θ|ni, so under the time reversal the wave function ψ(x0 ) = hx0 |αi Θ|p0 i = | − p0 i. The momentum space wave function transform thus under time reversal like φ(p0 ) −→ φ∗ (−p0 ). transforms like ψ(x0 ) −→ ψ ∗ (x0 ). If in particular we have ψ(x0 ) = R(r)Ylm (θ, φ), We consider a spin 21 particle the spin of which is oriented along n̂. The corresponding state is obtained by rotating the state |Sz ; ↑i: |n; ↑i = e−iSz α/h̄ e−iSy β/h̄ |Sz ; ↑i, we see that Ylm (θ, φ) −→ Ylm ∗ (θ, φ) = (−1)m Yl−m (θ, φ). Because Ylm is the wave function belonging to the state |lmi we must have m Θ|lmi = (−1) |l, −mi. The probability current corresponding to the wave function R(r)Ylm seems to turn clockwise when looked at from the direction of the positive z-axis and m > 0. The probability current of the corresponding time reversed state on the other hand turns counterclockwise because m changes its sign under the operation. The spinles particles obey Theorem 1 If the Hamiltonian H is invariant under the time reversal and the energy eigenstate |ni nondegenerate then the corresponding energy eigenfunction is real (or more generally a real function times a phase factor independent on the coordinate x0 ). where α and β are the direction angles of the vector n̂. Because ΘJ Θ−1 = −J . we see that Θ|n; ↑i = e−iSz α/h̄ e−iSy β/h̄ Θ|Sz ; ↑i. Furthermore, due to the oddity of the angular momentum, it follows that h̄ Jz Θ|Sz ; ↑i = − Θ|Sz ; ↑i, 2 so we must have Θ|Sz ; ↑i = η|Sz ; ↓i, where η is an arbitrary phase factor. So we get Θ|n; ↑i = η|n; ↓i. On the other hand we have Thus we must have |n; ↓i = e−iαSz /h̄ e−i(π+β)Sy /h̄ |Sz ; ↑i, Θ2 = (−1)2j . so Often one chooses −iSz α/h̄ −iSy β/h̄ η|n; ↓i = Θ|n; ↑i = e e Θ|Sz ; ↑i −iαSz /h̄ −i(π+β)Sy /h̄ = ηe e |Sz ; ↑i. Writing Θ|jmi = i2m |j, −mi. Spherical tensors Θ = U K, U unitary and recalling that the complex conjugation K has no effect on the base states we see that 2Sy K. Θ = ηe−iπSy /h̄ K = −iη h̄ Let us suppose that the operator A is either even or odd, i.e. ΘAΘ(−1) = ±A. We saw that then we have hα|A|αi = ±hα̃|A|α̃i. Now In an eigenstate of the angular momentum we have thus e−iπSy /h̄ |Sz ; ↑i = +|Sz ; ↓i e−iπSy /h̄ |Sz ; ↓i = −|Sz ; ↑i, so the effect of the time reversal on a general spin is hα, jm|A|α, jmi = ±hα, j, −m|A|α, j, −mi. 1 2 state Θ(c↑ |Sz ; ↑i + c↓ |Sz ; ↓i) = +ηc∗↑ |Sz ; ↓i − ηc∗↓ |Sz ; ↑i. Applying the operator Θ once again we get Θ2 (c↑ |Sz ; ↑i + c↓ |Sz ; ↓i) = −|η|2 c↑ |Sz ; ↑i − |η|2 c↓ |Sz ; ↓i = −(c↑ |Sz ; ↑i + c↓ |Sz ; ↓i), Let now A be a component of a Hermitian spherical tensor: A = Tq(k) . According to the Wigner-Eckart theorem it is sufficient to consider only the component q = 0. We define T (k) to be even/odd under the time reversal if (k) (k) ΘTq=0 Θ−1 = ±Tq=0 . Then we have i.e. for an arbitrary spin orientation we have (k) (k) hα, jm|T0 |α, jmi = ±hα, j, −m|T0 |α, j, −mi. 2 Θ = −1. From the relation Θ|lmi = (−1)m |l, −mi we see that for spinles particles we have The state |α, j, −mi is obtained by rotating the state |α, jmi: D(0, π, 0)|α, jmi = eiϕ |α, j, −mi. On the other hand, due to the definition of the spherical tensor Θ2 = 1. D † In general, one can show that Θ2 |j half integeri = −|j half integeri Θ2 |j integeri = +|j integeri. (R)Tq(k) D(R) = k X (k)∗ (k) Dqq0 (R)Tq0 , q 0 =−k we get (k) D† (0, π, 0)T0 D(0, π, 0) = X Generally we can write (k) D0q (0, π, 0)Tq(k) . q Θ = ηe−iπJy /h̄ K. Now (k) D00 (0, π, 0) = Pk (cos π) = (−1)k , Now e−2iπJy /h̄ |jmi = (−1)2j |jmi, so so we have (k) D† (0, π, 0)T0 D(0, π, 0) Θ2 |jmi = Θ ηe−iπJy /h̄ |jmi = |η|2 e−2iπJy /h̄ |jmi = (−1)2j |jmi. (k) = (−1)k T0 + (q 6= 0 components). Furthermore (k) hα, jm|Tq6=0 |α, jmi = 0, since the m selection rule would require m = m + q. So we get the Hamiltonian of an electron contains such terms as S · B, p · A + A · p. (k) hα, jm|T0 |α, jmi (k) = ±hα, jm|D† (0, π, 0)T0 D(0, π, 0)|α, jmi (k) = ±(−1)k hα, jm|T0 |α, jmi. Note Unlike under other symmetries the invariance of The magnetic field B is external, independent on the system, so [Θ, B] = 0 ja [Θ, A] = 0. On the other hand, S and p are odd, or the Hamiltonian under the time reversal [Θ, H] = 0, does not lead to any conservation laws. This is due to the fact that the time evolution operator is not invariant: ΘU (t, t0 ) 6= U (t, t0 )Θ. Time reversal and degeneracy Let us suppose that [Θ, H] = 0. Then the energy eigenstates obey H|ni = En |ni HΘ|ni = En Θ|ni. If we now had Θ|ni = eiδ |ni, then, reapplying the time reversal we would obtain Θ2 |ni = e−iδ Θ|ni = |ni, or Θ2 = 1. This is, however, impossible if the system j is half integer, because then Θ2 = −1. In systems of this kind |ni and Θ|ni are degenerate. Example Electon in electromagnetic field If a particle is influenced by an external static electric field V (x) = eφ(x), then clearly the Hamiltonian H= p2 + V (x) 2m is invariant under the time reversal: [Θ, H] = 0. If now there are odd number of electrons in the system the total j is half integer. Thus, in a system of this kind there is at least twofold degeneracy, so called Kramers’ degeneracy. In the magnetic field B =∇×A ΘSΘ−1 = −S ja ΘpΘ−1 = −p, so [Θ, H] 6= 0. We say that magnetic field breaks the time reversal symmetry and lifts the Kramers degeneracy.